Stage 9 - Q1 - 2 marks
A bag has 3 red and 3 blue counters. Draw twice, replacing and mixing after the first draw. Are the two red events independent?
Hint
Compare the second-draw probabilities after each possible first colour.
Stage 9 - Q2 - 2 marks
A fair coin has produced 12 heads in successive independent tosses. What is \(P(\text{tail next})\)? Does the streak change it?
Hint
Independence means the past result does not alter the next probability.
Stage 9 - Q3 - 2 marks
A bag has 6 red and 8 blue counters. Draw twice without replacement. Are the two red events independent?
Hint
Recount the bag after each possible first colour.
Stage 9 - Q4 - 3 marks
Choose uniformly from 1 to 16. A means even and B means at most 9. Test independence.
Hint
Count B separately among even and odd outcomes.
Stage 9 - Q5 - 3 marks
Red cards are numbered 1–8; blue cards 1–7. Choose one card uniformly. Are colour and evenness independent?
Hint
Compare even fractions within each colour, not just even counts.
Stage 9 - Q6 - 3 marks
On one uniform draw from 14 tokens, 4 are red and 10 blue. Red and blue are mutually exclusive. Are they independent?
Hint
Compare the actual joint probability with the product.
Stage 9 - Q7 - 3 marks
Choose uniformly from 1 to 18. A means even and B means at most 4. Test independence.
Hint
Count B separately among even and odd outcomes.
Stage 9 - Q8 - 2 marks
A bag has 3 red and 7 blue counters. Draw twice without replacement. Are the two red events independent?
Hint
Recount the bag after each possible first colour.
Stage 9 - Q9 - 3 marks
Red cards are numbered 1–6; blue cards 1–5. Choose one card uniformly. Are colour and evenness independent?
Hint
Compare even fractions within each colour, not just even counts.
Stage 9 - Q10 - 5 marks
Choose uniformly from 1 to 12. \(A=\{1,2,3,4\}\). Construct an event B with six outcomes that is independent of A. Justify your example.
Hint
Choose two outcomes from A and four from outside A.